21 research outputs found
Eigenvalue bounds of mixed Steklov problems
We study bounds on the Riesz means of the mixed Steklov-Neumann and
Steklov-Dirichlet eigenvalue problem on a bounded domain in
. The Steklov-Neumann eigenvalue problem is also called the
sloshing problem. We obtain two-term asymptotically sharp lower bounds on the
Riesz means of the sloshing problem and also provide an asymptotically sharp
upper bound for the Riesz means of mixed Steklov-Dirichlet problem. The proof
of our results for the sloshing problem uses the average variational principle
and monotonicity of sloshing eigenvalues. In the case of Steklov-Dirichlet
eigenvalue problem, the proof is based on a well-known bound on the Riesz means
of the Dirichlet fractional Laplacian and an inequality between the Dirichlet
and Navier fractional Laplacian. The two-term asymptotic results for the Riesz
means of mixed Steklov eigenvalue problems are discussed in the appendix which
in particular show the asymptotic sharpness of the bounds we obtain.Comment: An appendix by by F. Ferrulli and J. Lagac\'e is added; some changes
in the introduction are mad
The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Given two compact Riemannian manifolds with boundary and such
that their respective boundaries and admit neighborhoods
and which are isometric, we prove the existence of a
constant , which depends only on the geometry of ,
such that for each . This
follows from a quantitative relationship between the Steklov eigenvalues
of a compact Riemannian manifold and the eigenvalues
of the Laplacian on its boundary. Our main result states that the difference
is bounded above by a constant which depends on
the geometry of only in a neighborhood of its boundary. The proofs are
based on a Pohozaev identity and on comparison geometry for principal
curvatures of parallel hypersurfaces. In several situations, the constant
is given explicitly in terms of bounds on the geometry of
.Comment: 31 pages, 1 figur
On Pleijel's nodal domain theorem for the Robin problem
We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue
problem. In particular we remove the restriction, imposed in previous work,
that the Robin parameter be non-negative. We also improve the upper bound in
the statement of the Pleijel theorem. In the particular example of a Euclidean
ball, we calculate the explicit value of the Pleijel constant for a generic
constant Robin parameter and we show that it is equal to the Pleijel constant
for the Dirichlet Laplacian on a Euclidean ball.Comment: 17 page
Eigenvalues of the Laplacian and extrinsic geometry
We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space C P N instead of submanifolds of R N and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalu
Eigenvalues of the Laplacian and extrinsic geometry
We extend the results given by Colbois, Dryden and El Soufi on the
relationships between the eigenvalues of the Laplacian and an extrinsic
invariant called intersection index, in two directions. First, we replace this
intersection index by invariants of the same nature which are stable under
small perturbations. Second, we consider complex submanifolds of the complex
projective space instead of submanifolds of and
we obtain an eigenvalue upper bound depending only on the dimension of the
submanifold which is sharp for the first non-zero eigenvalue